Exactness of quadrature formulas

Journal article

The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We review the subject from this point of view and show that this principle fails to predict the actual behavior in four of the best-known cases: Newton–Cotes, Clenshaw–Curtis, Gauss–Legendre, and Gauss–Hermite quadrature. New results include (i) the observation that $x^k$ is integrated accurately by the Newton–Cotes formula even though the Chebyshev polynomial $T_k(x)$ is not; (ii) the introduction of a parameter-free variant of band-limited quadrature for arbitrary integrands, which is demonstrated to have a factor $\pi/2$ advantage over Gauss quadrature in integrating complex exponentials; (iii) a theorem establishing that chopping the real line to a finite interval achieves $O(\exp(-Cn^{2/3}))$ convergence for $n$-point quadrature of Gauss–Hermite integrands, whereas for the Gauss–Hermite formula it is just $O(\exp(-Cn^{1/2}))$; and (iv) an explanation of how this result is consistent with the “optimality” of the Gauss–Hermite formula.

Authors: Trefethen, LN

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Review
• Volume: 64
• Issue: 1
• Pages: 132–150
• Publication date: 2022-02-03
• Acceptance date: 2021-06-07
• DOI: 10.1137/20M1389522
• EISSN: 1095-7200
• ISSN: 0036-1445
• Language: English
• Keywords: FFR; Clenshaw–Curtis; Gauss quadrature; cubature; Gauss–Hermite; Newton–Cotes